A block of mass $m$ is suspended separately by two different springs have time period $t_1$ and $t_2$ . If same mass is connected to parallel combination of both springs, then its time period will be
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$\mathrm{t}=2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{k}}} \quad \Rightarrow \mathrm{k}=\frac{4 \pi^{2} \mathrm{m}}{\mathrm{t}^{2}}$
$\therefore \mathrm{k}_{1}=\frac{4 \pi^{2} \mathrm{m}}{\mathrm{t}_{1}^{2}}$ and $\mathrm{k}_{2}=\frac{4 \pi^{2} \mathrm{m}}{\mathrm{t}_{2}^{2}}$
$\mathrm{t}=2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{k}_{1}+\mathrm{k}_{2}}}=2 \pi \sqrt{\frac{\mathrm{m}}{\frac{4 \pi^{2} \mathrm{m}}{\mathrm{t}_{1}^{2}}+\frac{4 \pi^{2} \mathrm{m}}{\mathrm{t}_{2}^{2}}}}$
$\Rightarrow t=\frac{t_{1} t_{2}}{\sqrt{t_{1}^{2}+t_{2}^{2}}}$
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